Electrical Resistivity and Conductivity - Tensor Equations For Anisotropic Materials

Tensor Equations For Anisotropic Materials

Some materials are anisotropic, meaning they have different properties in different directions. For example, a crystal of graphite consists microscopically of a stack of sheets, and current flows very easily through each sheet, but moves much less easily from one sheet to the next.

For an anisotropic material, it is not generally valid to use the scalar equations

For example, the current may not flow in exactly the same direction as the electric field. Instead, the equations are generalized to the 3D tensor form

where the conductivity σ and resistivity ρ are rank-2 tensors (in other words, 3×3 matrices). The equations are compactly illustrated in component form (using index notation and the summation convention):

The σ and ρ tensors are inverses (in the sense of a matrix inverse). The individual components are not necessarily inverses; for example σ_xx may not be equal to 1/ρ_xx.